Multi Task Bayesian Compressed Sensing in Sparse 2D Spectroscopy

Introduction: J-coupling causes spectral splitting and complicated signal modulation that limit the detection of important brain metabolites, such as Glu, in proton spectroscopic imaging. While 2D spectroscopy, e.g. 2DJPRESS [1] and CTPRESS [2], has been demonstrated to successfully improve signal detection of coupled spins, it carries a penalty in scan time and reconstruction complexity. To counter this limitation, Mayer et al [3] exploited the diagonal feature of CTPRESS spectra to achieve four-fold undersampling without adverse aliasing artifacts. Exploration of further undersampling in 2D spectroscopy via compressed sensing appears promising as 2D spectra are naturally sparse and data sampling along the t1 encoding direction readily accommodates flexible sampling patterns. Here we modeled metabolite spectra for an under-sampled, noisy 2D CTPRESS spectroscopy at 3T, and evaluated the performance of multi-task Bayesian CS [4,5] which incorporated priors for regularization during reconstruction and compared it with Lustig's [6] implementation of conjugate gradient CS and single-task Bayesian CS [7]. Methods Using SPINEVOLUTION [8], 7 brain metabolites [9], (10 mM NAA, 7.9 mM Cr, 1.6 mM Cho, 9.2mM Glu, 4.5mM Gln, 6mM myo-Inositol, 0.4mM Lac) were simulated in a uniformly under sampled, 32-t1 step CTPRESS experiment with non-interfering aliasing as proposed by Mayer et al [3]. This 32-step 2D experiment is considered the baseline for further undersampling in this study, and was undersampled in the t1 dimension by factor R as determined by a random draw from a uniform distribution. Gaussian noise was added such that SNRNAA = 15 at full sampling. Reconstruction of the 2D spectra was obtained via three methods: i) CS via the non-linear conjugate-gradient solution [6], ii) Single-Task Bayesian CS [7], and iii) Multi-Task Bayesian CS [4]. The nonlinear conjugate-gradient solution is reproduced as Eq. 1 where y contains under sampled data, Φ is the sparse Fourier Transform , and m is the reconstructed data. λ is chosen as a balance between measurement consistency and enforced sparsity. In the joint Bayesian CS (Eqs. 2, 3 and 4) yi’s represent the under-sampled complex data, and fully sampled individual metabolite magnitude spectra as basis functions. The magnitude spectra were used as basis functions to approximate scanning conditions where phase priors are uncertain. α0 and A are the priors placed across all the spectra, and μi is the mean of posterior distribution for mi and is taken as its best estimate. The log-likelihood expression for α0 and A is conditioned upon all the yi’s and the maximization of this expression leads to evaluating μi and Σi. In single-task Bayesian CS, the expression for α0 and A is conditioned only on the under sampled spectra. Results and Discussion: Fig 2 and 3 show the reconstructed 2D spectra and corresponding 1D diagonal spectra for λ= 0.05 and R = 2 and R =3. At R = 2, the three CS methods restore NAA, Cr, Cho peaks in the diagonal spectra that were obscured in the zero-filled reconstruction. In addition Glu peaks were visible in the multi-task Bayesian CS reconstruction. At higher acceleration of R = 3, only NAA peaks were visible in the conjugate gradient CS reconstruction. Multi-task Bayesian CS reconstruction benefited from the prior information of fully sampled metabolite basis spectra, and recovered NAA, Cr, Cho, Glu peaks successfully. References: : [1] Hurd R et al; MRM 2004; 51:435-440. [2] Dreher et al; MRI 1999; 17:141-150 [3] Mayer D et al; MRM 2005; 54:439–442 [4] Ji et al; IEEE Trans. Sig. Proc 2009; 57;92-106 [5] Bilgic et al; ISMRM 2011 [6] Lustig M et al; MRM 2007; 58:1182-1195 [7] Ji et al; IEEE Trans. Sig. Proc 2008; 56;2346-2356 [8] Veshtort M et al; JMR 2006; 178:248-282 [9] Traber FB et al; JMRI 2004;19:537–545.